# Eigen-energies and eigenstates of a tri-atomic system

• Rafid Khanna
In summary: The eigen-energies of a system can be found by solving the Hamiltonian and the overlap matrix. In order to find the eigen-energies, you must assume that the three atoms are identical. This is done by assuming that the electron has the same eigen-energy for being in any of the three atoms.

## Homework Statement

An extra electron is added to one atom of a tri-atomic molecule. The electron has equal probability to jump to either of the other two atoms.

(a) Find the eigen-energies for the system. Assume that the new electron energy ##\bar{E_{0}}## is close to the non-hopping case energy ##E_{0}##. Draw an energy level diagram.

(b) Find one normalized eigenstate for the system.

## The Attempt at a Solution

(a) The only information available to me are that the the electron has equal probability to jump to either of the other two atoms. and the system has three atoms. How can these two pieces of information be used to find the eigen-energies of the system? Am I missing something?

I suppose you can assume that the three atoms are identical.

How might that help me? For now, all I can say is that the electron has the same eigen-energy for being in any of the three atoms i.e. in anyone of the three states.

Is that all I can say for the eigen-energies of the system?

I would also say that the model is somewhat ill specified. Did you do define "hopping" in class, maybe in connection with the Hubbard or Hueckel model? I think what you are supposed to assume is that the electron can be in one specific orbital on the atom, which is identical for all 3 atoms. Equal hopping probability translates into equal hamiltonian and overlap matrix elements between the orbitals. You then can set up some 3x3 Hamiltonian and Overlap matrix and try to solve it.

I think I will have to refer back to my QM textbooks and learn more about related topics before I can tackle this problem.

Can you please mention the topics that I must learn and be familiar with before I can answer this question?

## 1. What are eigen-energies and eigenstates of a tri-atomic system?

Eigen-energies and eigenstates are properties of a quantum system that describe the energy levels and corresponding wavefunctions of the system. In a tri-atomic system, there are three atoms interacting with each other, resulting in unique eigen-energies and eigenstates.

## 2. How are eigen-energies and eigenstates calculated for a tri-atomic system?

Eigen-energies and eigenstates are calculated using mathematical methods such as the Schrödinger equation or matrix diagonalization. These methods take into account the potential energy of the system and the interactions between the atoms to determine the energy levels and corresponding wavefunctions.

## 3. What is the significance of eigen-energies and eigenstates in a tri-atomic system?

Eigen-energies and eigenstates are important because they provide information about the behavior and properties of the system. They can help predict how the system will evolve over time and provide insights into the energy distribution and stability of the system.

## 4. Can eigen-energies and eigenstates change in a tri-atomic system?

Yes, eigen-energies and eigenstates can change if there are external factors that affect the system's potential energy or interactions between the atoms. These changes can alter the energy levels and corresponding wavefunctions of the system.

## 5. How do eigen-energies and eigenstates of a tri-atomic system relate to its molecular structure?

The eigen-energies and eigenstates of a tri-atomic system are closely related to its molecular structure. The specific arrangement and bonding between the atoms will affect the potential energy and interactions, resulting in unique eigen-energies and eigenstates for each molecule.